Properties

Label 161.2.a.b.1.1
Level 161
Weight 2
Character 161.1
Self dual yes
Analytic conductor 1.286
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 161.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.28559147254\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 161.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.23607 q^{5} +1.61803 q^{6} -1.00000 q^{7} +2.23607 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.23607 q^{5} +1.61803 q^{6} -1.00000 q^{7} +2.23607 q^{8} -2.00000 q^{9} -2.00000 q^{10} -4.47214 q^{11} -0.618034 q^{12} -4.23607 q^{13} +1.61803 q^{14} -1.23607 q^{15} -4.85410 q^{16} +3.23607 q^{18} -2.76393 q^{19} +0.763932 q^{20} +1.00000 q^{21} +7.23607 q^{22} -1.00000 q^{23} -2.23607 q^{24} -3.47214 q^{25} +6.85410 q^{26} +5.00000 q^{27} -0.618034 q^{28} +7.47214 q^{29} +2.00000 q^{30} -9.00000 q^{31} +3.38197 q^{32} +4.47214 q^{33} -1.23607 q^{35} -1.23607 q^{36} +7.70820 q^{37} +4.47214 q^{38} +4.23607 q^{39} +2.76393 q^{40} +2.23607 q^{41} -1.61803 q^{42} -6.47214 q^{43} -2.76393 q^{44} -2.47214 q^{45} +1.61803 q^{46} +5.47214 q^{47} +4.85410 q^{48} +1.00000 q^{49} +5.61803 q^{50} -2.61803 q^{52} +6.76393 q^{53} -8.09017 q^{54} -5.52786 q^{55} -2.23607 q^{56} +2.76393 q^{57} -12.0902 q^{58} -10.4721 q^{59} -0.763932 q^{60} -13.4164 q^{61} +14.5623 q^{62} +2.00000 q^{63} +4.23607 q^{64} -5.23607 q^{65} -7.23607 q^{66} +10.1803 q^{67} +1.00000 q^{69} +2.00000 q^{70} -5.76393 q^{71} -4.47214 q^{72} +6.70820 q^{73} -12.4721 q^{74} +3.47214 q^{75} -1.70820 q^{76} +4.47214 q^{77} -6.85410 q^{78} -2.76393 q^{79} -6.00000 q^{80} +1.00000 q^{81} -3.61803 q^{82} -2.47214 q^{83} +0.618034 q^{84} +10.4721 q^{86} -7.47214 q^{87} -10.0000 q^{88} -8.94427 q^{89} +4.00000 q^{90} +4.23607 q^{91} -0.618034 q^{92} +9.00000 q^{93} -8.85410 q^{94} -3.41641 q^{95} -3.38197 q^{96} -9.70820 q^{97} -1.61803 q^{98} +8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} - 4q^{9} + O(q^{10}) \) \( 2q - q^{2} - 2q^{3} - q^{4} - 2q^{5} + q^{6} - 2q^{7} - 4q^{9} - 4q^{10} + q^{12} - 4q^{13} + q^{14} + 2q^{15} - 3q^{16} + 2q^{18} - 10q^{19} + 6q^{20} + 2q^{21} + 10q^{22} - 2q^{23} + 2q^{25} + 7q^{26} + 10q^{27} + q^{28} + 6q^{29} + 4q^{30} - 18q^{31} + 9q^{32} + 2q^{35} + 2q^{36} + 2q^{37} + 4q^{39} + 10q^{40} - q^{42} - 4q^{43} - 10q^{44} + 4q^{45} + q^{46} + 2q^{47} + 3q^{48} + 2q^{49} + 9q^{50} - 3q^{52} + 18q^{53} - 5q^{54} - 20q^{55} + 10q^{57} - 13q^{58} - 12q^{59} - 6q^{60} + 9q^{62} + 4q^{63} + 4q^{64} - 6q^{65} - 10q^{66} - 2q^{67} + 2q^{69} + 4q^{70} - 16q^{71} - 16q^{74} - 2q^{75} + 10q^{76} - 7q^{78} - 10q^{79} - 12q^{80} + 2q^{81} - 5q^{82} + 4q^{83} - q^{84} + 12q^{86} - 6q^{87} - 20q^{88} + 8q^{90} + 4q^{91} + q^{92} + 18q^{93} - 11q^{94} + 20q^{95} - 9q^{96} - 6q^{97} - q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0.618034 0.309017
\(5\) 1.23607 0.552786 0.276393 0.961045i \(-0.410861\pi\)
0.276393 + 0.961045i \(0.410861\pi\)
\(6\) 1.61803 0.660560
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) −2.00000 −0.666667
\(10\) −2.00000 −0.632456
\(11\) −4.47214 −1.34840 −0.674200 0.738549i \(-0.735511\pi\)
−0.674200 + 0.738549i \(0.735511\pi\)
\(12\) −0.618034 −0.178411
\(13\) −4.23607 −1.17487 −0.587437 0.809270i \(-0.699863\pi\)
−0.587437 + 0.809270i \(0.699863\pi\)
\(14\) 1.61803 0.432438
\(15\) −1.23607 −0.319151
\(16\) −4.85410 −1.21353
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 3.23607 0.762749
\(19\) −2.76393 −0.634089 −0.317045 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 0.763932 0.170820
\(21\) 1.00000 0.218218
\(22\) 7.23607 1.54273
\(23\) −1.00000 −0.208514
\(24\) −2.23607 −0.456435
\(25\) −3.47214 −0.694427
\(26\) 6.85410 1.34420
\(27\) 5.00000 0.962250
\(28\) −0.618034 −0.116797
\(29\) 7.47214 1.38754 0.693770 0.720196i \(-0.255948\pi\)
0.693770 + 0.720196i \(0.255948\pi\)
\(30\) 2.00000 0.365148
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 3.38197 0.597853
\(33\) 4.47214 0.778499
\(34\) 0 0
\(35\) −1.23607 −0.208934
\(36\) −1.23607 −0.206011
\(37\) 7.70820 1.26722 0.633610 0.773652i \(-0.281572\pi\)
0.633610 + 0.773652i \(0.281572\pi\)
\(38\) 4.47214 0.725476
\(39\) 4.23607 0.678314
\(40\) 2.76393 0.437016
\(41\) 2.23607 0.349215 0.174608 0.984638i \(-0.444134\pi\)
0.174608 + 0.984638i \(0.444134\pi\)
\(42\) −1.61803 −0.249668
\(43\) −6.47214 −0.986991 −0.493496 0.869748i \(-0.664281\pi\)
−0.493496 + 0.869748i \(0.664281\pi\)
\(44\) −2.76393 −0.416678
\(45\) −2.47214 −0.368524
\(46\) 1.61803 0.238566
\(47\) 5.47214 0.798193 0.399097 0.916909i \(-0.369324\pi\)
0.399097 + 0.916909i \(0.369324\pi\)
\(48\) 4.85410 0.700629
\(49\) 1.00000 0.142857
\(50\) 5.61803 0.794510
\(51\) 0 0
\(52\) −2.61803 −0.363056
\(53\) 6.76393 0.929098 0.464549 0.885548i \(-0.346217\pi\)
0.464549 + 0.885548i \(0.346217\pi\)
\(54\) −8.09017 −1.10093
\(55\) −5.52786 −0.745377
\(56\) −2.23607 −0.298807
\(57\) 2.76393 0.366092
\(58\) −12.0902 −1.58752
\(59\) −10.4721 −1.36336 −0.681678 0.731652i \(-0.738749\pi\)
−0.681678 + 0.731652i \(0.738749\pi\)
\(60\) −0.763932 −0.0986232
\(61\) −13.4164 −1.71780 −0.858898 0.512148i \(-0.828850\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 14.5623 1.84941
\(63\) 2.00000 0.251976
\(64\) 4.23607 0.529508
\(65\) −5.23607 −0.649454
\(66\) −7.23607 −0.890698
\(67\) 10.1803 1.24373 0.621863 0.783126i \(-0.286376\pi\)
0.621863 + 0.783126i \(0.286376\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 2.00000 0.239046
\(71\) −5.76393 −0.684053 −0.342026 0.939690i \(-0.611113\pi\)
−0.342026 + 0.939690i \(0.611113\pi\)
\(72\) −4.47214 −0.527046
\(73\) 6.70820 0.785136 0.392568 0.919723i \(-0.371587\pi\)
0.392568 + 0.919723i \(0.371587\pi\)
\(74\) −12.4721 −1.44986
\(75\) 3.47214 0.400928
\(76\) −1.70820 −0.195944
\(77\) 4.47214 0.509647
\(78\) −6.85410 −0.776074
\(79\) −2.76393 −0.310967 −0.155483 0.987839i \(-0.549693\pi\)
−0.155483 + 0.987839i \(0.549693\pi\)
\(80\) −6.00000 −0.670820
\(81\) 1.00000 0.111111
\(82\) −3.61803 −0.399545
\(83\) −2.47214 −0.271352 −0.135676 0.990753i \(-0.543321\pi\)
−0.135676 + 0.990753i \(0.543321\pi\)
\(84\) 0.618034 0.0674330
\(85\) 0 0
\(86\) 10.4721 1.12924
\(87\) −7.47214 −0.801097
\(88\) −10.0000 −1.06600
\(89\) −8.94427 −0.948091 −0.474045 0.880500i \(-0.657207\pi\)
−0.474045 + 0.880500i \(0.657207\pi\)
\(90\) 4.00000 0.421637
\(91\) 4.23607 0.444061
\(92\) −0.618034 −0.0644345
\(93\) 9.00000 0.933257
\(94\) −8.85410 −0.913231
\(95\) −3.41641 −0.350516
\(96\) −3.38197 −0.345170
\(97\) −9.70820 −0.985719 −0.492859 0.870109i \(-0.664048\pi\)
−0.492859 + 0.870109i \(0.664048\pi\)
\(98\) −1.61803 −0.163446
\(99\) 8.94427 0.898933
\(100\) −2.14590 −0.214590
\(101\) 13.4164 1.33498 0.667491 0.744618i \(-0.267368\pi\)
0.667491 + 0.744618i \(0.267368\pi\)
\(102\) 0 0
\(103\) −2.29180 −0.225817 −0.112909 0.993605i \(-0.536017\pi\)
−0.112909 + 0.993605i \(0.536017\pi\)
\(104\) −9.47214 −0.928819
\(105\) 1.23607 0.120628
\(106\) −10.9443 −1.06300
\(107\) 6.76393 0.653894 0.326947 0.945043i \(-0.393980\pi\)
0.326947 + 0.945043i \(0.393980\pi\)
\(108\) 3.09017 0.297352
\(109\) 11.4164 1.09349 0.546747 0.837298i \(-0.315866\pi\)
0.546747 + 0.837298i \(0.315866\pi\)
\(110\) 8.94427 0.852803
\(111\) −7.70820 −0.731630
\(112\) 4.85410 0.458670
\(113\) −6.47214 −0.608847 −0.304424 0.952537i \(-0.598464\pi\)
−0.304424 + 0.952537i \(0.598464\pi\)
\(114\) −4.47214 −0.418854
\(115\) −1.23607 −0.115264
\(116\) 4.61803 0.428774
\(117\) 8.47214 0.783249
\(118\) 16.9443 1.55985
\(119\) 0 0
\(120\) −2.76393 −0.252311
\(121\) 9.00000 0.818182
\(122\) 21.7082 1.96537
\(123\) −2.23607 −0.201619
\(124\) −5.56231 −0.499510
\(125\) −10.4721 −0.936656
\(126\) −3.23607 −0.288292
\(127\) 10.7082 0.950199 0.475100 0.879932i \(-0.342412\pi\)
0.475100 + 0.879932i \(0.342412\pi\)
\(128\) −13.6180 −1.20368
\(129\) 6.47214 0.569840
\(130\) 8.47214 0.743055
\(131\) −13.9443 −1.21832 −0.609158 0.793049i \(-0.708493\pi\)
−0.609158 + 0.793049i \(0.708493\pi\)
\(132\) 2.76393 0.240569
\(133\) 2.76393 0.239663
\(134\) −16.4721 −1.42298
\(135\) 6.18034 0.531919
\(136\) 0 0
\(137\) 2.29180 0.195801 0.0979007 0.995196i \(-0.468787\pi\)
0.0979007 + 0.995196i \(0.468787\pi\)
\(138\) −1.61803 −0.137736
\(139\) 11.4721 0.973054 0.486527 0.873666i \(-0.338264\pi\)
0.486527 + 0.873666i \(0.338264\pi\)
\(140\) −0.763932 −0.0645640
\(141\) −5.47214 −0.460837
\(142\) 9.32624 0.782641
\(143\) 18.9443 1.58420
\(144\) 9.70820 0.809017
\(145\) 9.23607 0.767014
\(146\) −10.8541 −0.898292
\(147\) −1.00000 −0.0824786
\(148\) 4.76393 0.391593
\(149\) 10.7639 0.881816 0.440908 0.897552i \(-0.354657\pi\)
0.440908 + 0.897552i \(0.354657\pi\)
\(150\) −5.61803 −0.458711
\(151\) 7.18034 0.584328 0.292164 0.956368i \(-0.405625\pi\)
0.292164 + 0.956368i \(0.405625\pi\)
\(152\) −6.18034 −0.501292
\(153\) 0 0
\(154\) −7.23607 −0.583099
\(155\) −11.1246 −0.893550
\(156\) 2.61803 0.209610
\(157\) −11.4164 −0.911129 −0.455564 0.890203i \(-0.650562\pi\)
−0.455564 + 0.890203i \(0.650562\pi\)
\(158\) 4.47214 0.355784
\(159\) −6.76393 −0.536415
\(160\) 4.18034 0.330485
\(161\) 1.00000 0.0788110
\(162\) −1.61803 −0.127125
\(163\) −3.18034 −0.249103 −0.124552 0.992213i \(-0.539749\pi\)
−0.124552 + 0.992213i \(0.539749\pi\)
\(164\) 1.38197 0.107913
\(165\) 5.52786 0.430344
\(166\) 4.00000 0.310460
\(167\) −13.8885 −1.07473 −0.537364 0.843350i \(-0.680580\pi\)
−0.537364 + 0.843350i \(0.680580\pi\)
\(168\) 2.23607 0.172516
\(169\) 4.94427 0.380329
\(170\) 0 0
\(171\) 5.52786 0.422726
\(172\) −4.00000 −0.304997
\(173\) −12.4721 −0.948239 −0.474119 0.880461i \(-0.657234\pi\)
−0.474119 + 0.880461i \(0.657234\pi\)
\(174\) 12.0902 0.916553
\(175\) 3.47214 0.262469
\(176\) 21.7082 1.63632
\(177\) 10.4721 0.787134
\(178\) 14.4721 1.08473
\(179\) −5.29180 −0.395527 −0.197764 0.980250i \(-0.563368\pi\)
−0.197764 + 0.980250i \(0.563368\pi\)
\(180\) −1.52786 −0.113880
\(181\) −22.9443 −1.70543 −0.852717 0.522373i \(-0.825047\pi\)
−0.852717 + 0.522373i \(0.825047\pi\)
\(182\) −6.85410 −0.508060
\(183\) 13.4164 0.991769
\(184\) −2.23607 −0.164845
\(185\) 9.52786 0.700502
\(186\) −14.5623 −1.06776
\(187\) 0 0
\(188\) 3.38197 0.246655
\(189\) −5.00000 −0.363696
\(190\) 5.52786 0.401033
\(191\) −1.81966 −0.131666 −0.0658330 0.997831i \(-0.520970\pi\)
−0.0658330 + 0.997831i \(0.520970\pi\)
\(192\) −4.23607 −0.305712
\(193\) −22.4164 −1.61357 −0.806784 0.590846i \(-0.798794\pi\)
−0.806784 + 0.590846i \(0.798794\pi\)
\(194\) 15.7082 1.12778
\(195\) 5.23607 0.374963
\(196\) 0.618034 0.0441453
\(197\) −12.5279 −0.892573 −0.446287 0.894890i \(-0.647254\pi\)
−0.446287 + 0.894890i \(0.647254\pi\)
\(198\) −14.4721 −1.02849
\(199\) −19.8885 −1.40986 −0.704931 0.709276i \(-0.749022\pi\)
−0.704931 + 0.709276i \(0.749022\pi\)
\(200\) −7.76393 −0.548993
\(201\) −10.1803 −0.718066
\(202\) −21.7082 −1.52738
\(203\) −7.47214 −0.524441
\(204\) 0 0
\(205\) 2.76393 0.193041
\(206\) 3.70820 0.258363
\(207\) 2.00000 0.139010
\(208\) 20.5623 1.42574
\(209\) 12.3607 0.855006
\(210\) −2.00000 −0.138013
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 4.18034 0.287107
\(213\) 5.76393 0.394938
\(214\) −10.9443 −0.748135
\(215\) −8.00000 −0.545595
\(216\) 11.1803 0.760726
\(217\) 9.00000 0.610960
\(218\) −18.4721 −1.25109
\(219\) −6.70820 −0.453298
\(220\) −3.41641 −0.230334
\(221\) 0 0
\(222\) 12.4721 0.837075
\(223\) −23.4164 −1.56808 −0.784039 0.620711i \(-0.786844\pi\)
−0.784039 + 0.620711i \(0.786844\pi\)
\(224\) −3.38197 −0.225967
\(225\) 6.94427 0.462951
\(226\) 10.4721 0.696596
\(227\) 29.8885 1.98377 0.991886 0.127129i \(-0.0405763\pi\)
0.991886 + 0.127129i \(0.0405763\pi\)
\(228\) 1.70820 0.113129
\(229\) 19.2361 1.27116 0.635578 0.772037i \(-0.280762\pi\)
0.635578 + 0.772037i \(0.280762\pi\)
\(230\) 2.00000 0.131876
\(231\) −4.47214 −0.294245
\(232\) 16.7082 1.09695
\(233\) −2.52786 −0.165606 −0.0828029 0.996566i \(-0.526387\pi\)
−0.0828029 + 0.996566i \(0.526387\pi\)
\(234\) −13.7082 −0.896133
\(235\) 6.76393 0.441230
\(236\) −6.47214 −0.421300
\(237\) 2.76393 0.179537
\(238\) 0 0
\(239\) −20.2361 −1.30896 −0.654481 0.756078i \(-0.727113\pi\)
−0.654481 + 0.756078i \(0.727113\pi\)
\(240\) 6.00000 0.387298
\(241\) 5.41641 0.348902 0.174451 0.984666i \(-0.444185\pi\)
0.174451 + 0.984666i \(0.444185\pi\)
\(242\) −14.5623 −0.936100
\(243\) −16.0000 −1.02640
\(244\) −8.29180 −0.530828
\(245\) 1.23607 0.0789695
\(246\) 3.61803 0.230677
\(247\) 11.7082 0.744975
\(248\) −20.1246 −1.27791
\(249\) 2.47214 0.156665
\(250\) 16.9443 1.07165
\(251\) 21.7082 1.37021 0.685105 0.728444i \(-0.259756\pi\)
0.685105 + 0.728444i \(0.259756\pi\)
\(252\) 1.23607 0.0778650
\(253\) 4.47214 0.281161
\(254\) −17.3262 −1.08714
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) −0.236068 −0.0147255 −0.00736276 0.999973i \(-0.502344\pi\)
−0.00736276 + 0.999973i \(0.502344\pi\)
\(258\) −10.4721 −0.651967
\(259\) −7.70820 −0.478964
\(260\) −3.23607 −0.200692
\(261\) −14.9443 −0.925027
\(262\) 22.5623 1.39390
\(263\) −9.05573 −0.558400 −0.279200 0.960233i \(-0.590069\pi\)
−0.279200 + 0.960233i \(0.590069\pi\)
\(264\) 10.0000 0.615457
\(265\) 8.36068 0.513592
\(266\) −4.47214 −0.274204
\(267\) 8.94427 0.547381
\(268\) 6.29180 0.384333
\(269\) 13.1803 0.803620 0.401810 0.915723i \(-0.368381\pi\)
0.401810 + 0.915723i \(0.368381\pi\)
\(270\) −10.0000 −0.608581
\(271\) 0.944272 0.0573604 0.0286802 0.999589i \(-0.490870\pi\)
0.0286802 + 0.999589i \(0.490870\pi\)
\(272\) 0 0
\(273\) −4.23607 −0.256378
\(274\) −3.70820 −0.224021
\(275\) 15.5279 0.936365
\(276\) 0.618034 0.0372013
\(277\) −6.41641 −0.385525 −0.192762 0.981245i \(-0.561745\pi\)
−0.192762 + 0.981245i \(0.561745\pi\)
\(278\) −18.5623 −1.11329
\(279\) 18.0000 1.07763
\(280\) −2.76393 −0.165177
\(281\) −9.70820 −0.579143 −0.289571 0.957156i \(-0.593513\pi\)
−0.289571 + 0.957156i \(0.593513\pi\)
\(282\) 8.85410 0.527254
\(283\) 1.05573 0.0627565 0.0313783 0.999508i \(-0.490010\pi\)
0.0313783 + 0.999508i \(0.490010\pi\)
\(284\) −3.56231 −0.211384
\(285\) 3.41641 0.202371
\(286\) −30.6525 −1.81252
\(287\) −2.23607 −0.131991
\(288\) −6.76393 −0.398569
\(289\) −17.0000 −1.00000
\(290\) −14.9443 −0.877558
\(291\) 9.70820 0.569105
\(292\) 4.14590 0.242620
\(293\) −2.29180 −0.133888 −0.0669441 0.997757i \(-0.521325\pi\)
−0.0669441 + 0.997757i \(0.521325\pi\)
\(294\) 1.61803 0.0943657
\(295\) −12.9443 −0.753645
\(296\) 17.2361 1.00183
\(297\) −22.3607 −1.29750
\(298\) −17.4164 −1.00891
\(299\) 4.23607 0.244978
\(300\) 2.14590 0.123893
\(301\) 6.47214 0.373048
\(302\) −11.6180 −0.668543
\(303\) −13.4164 −0.770752
\(304\) 13.4164 0.769484
\(305\) −16.5836 −0.949574
\(306\) 0 0
\(307\) 15.4164 0.879861 0.439930 0.898032i \(-0.355003\pi\)
0.439930 + 0.898032i \(0.355003\pi\)
\(308\) 2.76393 0.157490
\(309\) 2.29180 0.130376
\(310\) 18.0000 1.02233
\(311\) 32.8885 1.86494 0.932469 0.361250i \(-0.117650\pi\)
0.932469 + 0.361250i \(0.117650\pi\)
\(312\) 9.47214 0.536254
\(313\) 7.23607 0.409007 0.204503 0.978866i \(-0.434442\pi\)
0.204503 + 0.978866i \(0.434442\pi\)
\(314\) 18.4721 1.04244
\(315\) 2.47214 0.139289
\(316\) −1.70820 −0.0960940
\(317\) −31.3050 −1.75826 −0.879131 0.476581i \(-0.841876\pi\)
−0.879131 + 0.476581i \(0.841876\pi\)
\(318\) 10.9443 0.613724
\(319\) −33.4164 −1.87096
\(320\) 5.23607 0.292705
\(321\) −6.76393 −0.377526
\(322\) −1.61803 −0.0901695
\(323\) 0 0
\(324\) 0.618034 0.0343352
\(325\) 14.7082 0.815864
\(326\) 5.14590 0.285005
\(327\) −11.4164 −0.631329
\(328\) 5.00000 0.276079
\(329\) −5.47214 −0.301689
\(330\) −8.94427 −0.492366
\(331\) −12.7082 −0.698506 −0.349253 0.937028i \(-0.613565\pi\)
−0.349253 + 0.937028i \(0.613565\pi\)
\(332\) −1.52786 −0.0838524
\(333\) −15.4164 −0.844814
\(334\) 22.4721 1.22962
\(335\) 12.5836 0.687515
\(336\) −4.85410 −0.264813
\(337\) 31.5967 1.72118 0.860592 0.509295i \(-0.170094\pi\)
0.860592 + 0.509295i \(0.170094\pi\)
\(338\) −8.00000 −0.435143
\(339\) 6.47214 0.351518
\(340\) 0 0
\(341\) 40.2492 2.17962
\(342\) −8.94427 −0.483651
\(343\) −1.00000 −0.0539949
\(344\) −14.4721 −0.780285
\(345\) 1.23607 0.0665477
\(346\) 20.1803 1.08490
\(347\) 29.8885 1.60450 0.802251 0.596987i \(-0.203636\pi\)
0.802251 + 0.596987i \(0.203636\pi\)
\(348\) −4.61803 −0.247553
\(349\) 14.7082 0.787312 0.393656 0.919258i \(-0.371210\pi\)
0.393656 + 0.919258i \(0.371210\pi\)
\(350\) −5.61803 −0.300297
\(351\) −21.1803 −1.13052
\(352\) −15.1246 −0.806145
\(353\) −6.23607 −0.331912 −0.165956 0.986133i \(-0.553071\pi\)
−0.165956 + 0.986133i \(0.553071\pi\)
\(354\) −16.9443 −0.900578
\(355\) −7.12461 −0.378135
\(356\) −5.52786 −0.292976
\(357\) 0 0
\(358\) 8.56231 0.452532
\(359\) 9.23607 0.487461 0.243731 0.969843i \(-0.421629\pi\)
0.243731 + 0.969843i \(0.421629\pi\)
\(360\) −5.52786 −0.291344
\(361\) −11.3607 −0.597931
\(362\) 37.1246 1.95123
\(363\) −9.00000 −0.472377
\(364\) 2.61803 0.137222
\(365\) 8.29180 0.434012
\(366\) −21.7082 −1.13471
\(367\) −31.4164 −1.63992 −0.819962 0.572419i \(-0.806005\pi\)
−0.819962 + 0.572419i \(0.806005\pi\)
\(368\) 4.85410 0.253038
\(369\) −4.47214 −0.232810
\(370\) −15.4164 −0.801461
\(371\) −6.76393 −0.351166
\(372\) 5.56231 0.288392
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 10.4721 0.540779
\(376\) 12.2361 0.631027
\(377\) −31.6525 −1.63019
\(378\) 8.09017 0.416113
\(379\) 21.7082 1.11508 0.557538 0.830152i \(-0.311746\pi\)
0.557538 + 0.830152i \(0.311746\pi\)
\(380\) −2.11146 −0.108315
\(381\) −10.7082 −0.548598
\(382\) 2.94427 0.150642
\(383\) −2.29180 −0.117105 −0.0585527 0.998284i \(-0.518649\pi\)
−0.0585527 + 0.998284i \(0.518649\pi\)
\(384\) 13.6180 0.694942
\(385\) 5.52786 0.281726
\(386\) 36.2705 1.84612
\(387\) 12.9443 0.657994
\(388\) −6.00000 −0.304604
\(389\) 19.4164 0.984451 0.492225 0.870468i \(-0.336184\pi\)
0.492225 + 0.870468i \(0.336184\pi\)
\(390\) −8.47214 −0.429003
\(391\) 0 0
\(392\) 2.23607 0.112938
\(393\) 13.9443 0.703395
\(394\) 20.2705 1.02121
\(395\) −3.41641 −0.171898
\(396\) 5.52786 0.277786
\(397\) 5.65248 0.283690 0.141845 0.989889i \(-0.454697\pi\)
0.141845 + 0.989889i \(0.454697\pi\)
\(398\) 32.1803 1.61305
\(399\) −2.76393 −0.138370
\(400\) 16.8541 0.842705
\(401\) 15.8885 0.793436 0.396718 0.917941i \(-0.370149\pi\)
0.396718 + 0.917941i \(0.370149\pi\)
\(402\) 16.4721 0.821555
\(403\) 38.1246 1.89912
\(404\) 8.29180 0.412532
\(405\) 1.23607 0.0614207
\(406\) 12.0902 0.600025
\(407\) −34.4721 −1.70872
\(408\) 0 0
\(409\) −36.1246 −1.78625 −0.893124 0.449811i \(-0.851491\pi\)
−0.893124 + 0.449811i \(0.851491\pi\)
\(410\) −4.47214 −0.220863
\(411\) −2.29180 −0.113046
\(412\) −1.41641 −0.0697814
\(413\) 10.4721 0.515300
\(414\) −3.23607 −0.159044
\(415\) −3.05573 −0.150000
\(416\) −14.3262 −0.702402
\(417\) −11.4721 −0.561793
\(418\) −20.0000 −0.978232
\(419\) 17.5279 0.856292 0.428146 0.903710i \(-0.359167\pi\)
0.428146 + 0.903710i \(0.359167\pi\)
\(420\) 0.763932 0.0372761
\(421\) −35.4164 −1.72609 −0.863045 0.505127i \(-0.831446\pi\)
−0.863045 + 0.505127i \(0.831446\pi\)
\(422\) 19.4164 0.945176
\(423\) −10.9443 −0.532129
\(424\) 15.1246 0.734516
\(425\) 0 0
\(426\) −9.32624 −0.451858
\(427\) 13.4164 0.649265
\(428\) 4.18034 0.202064
\(429\) −18.9443 −0.914638
\(430\) 12.9443 0.624228
\(431\) −4.18034 −0.201360 −0.100680 0.994919i \(-0.532102\pi\)
−0.100680 + 0.994919i \(0.532102\pi\)
\(432\) −24.2705 −1.16772
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −14.5623 −0.699013
\(435\) −9.23607 −0.442836
\(436\) 7.05573 0.337908
\(437\) 2.76393 0.132217
\(438\) 10.8541 0.518629
\(439\) −32.3050 −1.54183 −0.770916 0.636937i \(-0.780201\pi\)
−0.770916 + 0.636937i \(0.780201\pi\)
\(440\) −12.3607 −0.589272
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 15.1803 0.721240 0.360620 0.932713i \(-0.382565\pi\)
0.360620 + 0.932713i \(0.382565\pi\)
\(444\) −4.76393 −0.226086
\(445\) −11.0557 −0.524092
\(446\) 37.8885 1.79407
\(447\) −10.7639 −0.509117
\(448\) −4.23607 −0.200135
\(449\) −20.8328 −0.983161 −0.491581 0.870832i \(-0.663581\pi\)
−0.491581 + 0.870832i \(0.663581\pi\)
\(450\) −11.2361 −0.529673
\(451\) −10.0000 −0.470882
\(452\) −4.00000 −0.188144
\(453\) −7.18034 −0.337362
\(454\) −48.3607 −2.26968
\(455\) 5.23607 0.245471
\(456\) 6.18034 0.289421
\(457\) 5.52786 0.258583 0.129291 0.991607i \(-0.458730\pi\)
0.129291 + 0.991607i \(0.458730\pi\)
\(458\) −31.1246 −1.45436
\(459\) 0 0
\(460\) −0.763932 −0.0356185
\(461\) −30.2361 −1.40823 −0.704117 0.710084i \(-0.748657\pi\)
−0.704117 + 0.710084i \(0.748657\pi\)
\(462\) 7.23607 0.336652
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −36.2705 −1.68382
\(465\) 11.1246 0.515892
\(466\) 4.09017 0.189473
\(467\) −20.4721 −0.947337 −0.473669 0.880703i \(-0.657071\pi\)
−0.473669 + 0.880703i \(0.657071\pi\)
\(468\) 5.23607 0.242037
\(469\) −10.1803 −0.470084
\(470\) −10.9443 −0.504822
\(471\) 11.4164 0.526040
\(472\) −23.4164 −1.07783
\(473\) 28.9443 1.33086
\(474\) −4.47214 −0.205412
\(475\) 9.59675 0.440329
\(476\) 0 0
\(477\) −13.5279 −0.619398
\(478\) 32.7426 1.49761
\(479\) 17.1246 0.782443 0.391222 0.920296i \(-0.372053\pi\)
0.391222 + 0.920296i \(0.372053\pi\)
\(480\) −4.18034 −0.190806
\(481\) −32.6525 −1.48882
\(482\) −8.76393 −0.399186
\(483\) −1.00000 −0.0455016
\(484\) 5.56231 0.252832
\(485\) −12.0000 −0.544892
\(486\) 25.8885 1.17433
\(487\) −20.1246 −0.911933 −0.455967 0.889997i \(-0.650706\pi\)
−0.455967 + 0.889997i \(0.650706\pi\)
\(488\) −30.0000 −1.35804
\(489\) 3.18034 0.143820
\(490\) −2.00000 −0.0903508
\(491\) 16.2361 0.732723 0.366362 0.930472i \(-0.380603\pi\)
0.366362 + 0.930472i \(0.380603\pi\)
\(492\) −1.38197 −0.0623038
\(493\) 0 0
\(494\) −18.9443 −0.852343
\(495\) 11.0557 0.496918
\(496\) 43.6869 1.96160
\(497\) 5.76393 0.258548
\(498\) −4.00000 −0.179244
\(499\) −23.2918 −1.04268 −0.521342 0.853348i \(-0.674568\pi\)
−0.521342 + 0.853348i \(0.674568\pi\)
\(500\) −6.47214 −0.289443
\(501\) 13.8885 0.620494
\(502\) −35.1246 −1.56769
\(503\) 33.5967 1.49800 0.749002 0.662567i \(-0.230533\pi\)
0.749002 + 0.662567i \(0.230533\pi\)
\(504\) 4.47214 0.199205
\(505\) 16.5836 0.737960
\(506\) −7.23607 −0.321682
\(507\) −4.94427 −0.219583
\(508\) 6.61803 0.293628
\(509\) −35.1803 −1.55934 −0.779671 0.626190i \(-0.784613\pi\)
−0.779671 + 0.626190i \(0.784613\pi\)
\(510\) 0 0
\(511\) −6.70820 −0.296753
\(512\) 5.29180 0.233867
\(513\) −13.8197 −0.610153
\(514\) 0.381966 0.0168478
\(515\) −2.83282 −0.124829
\(516\) 4.00000 0.176090
\(517\) −24.4721 −1.07628
\(518\) 12.4721 0.547994
\(519\) 12.4721 0.547466
\(520\) −11.7082 −0.513439
\(521\) −41.3050 −1.80960 −0.904801 0.425834i \(-0.859981\pi\)
−0.904801 + 0.425834i \(0.859981\pi\)
\(522\) 24.1803 1.05834
\(523\) 5.70820 0.249602 0.124801 0.992182i \(-0.460171\pi\)
0.124801 + 0.992182i \(0.460171\pi\)
\(524\) −8.61803 −0.376481
\(525\) −3.47214 −0.151536
\(526\) 14.6525 0.638878
\(527\) 0 0
\(528\) −21.7082 −0.944728
\(529\) 1.00000 0.0434783
\(530\) −13.5279 −0.587613
\(531\) 20.9443 0.908904
\(532\) 1.70820 0.0740600
\(533\) −9.47214 −0.410284
\(534\) −14.4721 −0.626271
\(535\) 8.36068 0.361464
\(536\) 22.7639 0.983252
\(537\) 5.29180 0.228358
\(538\) −21.3262 −0.919439
\(539\) −4.47214 −0.192629
\(540\) 3.81966 0.164372
\(541\) 21.0000 0.902861 0.451430 0.892306i \(-0.350914\pi\)
0.451430 + 0.892306i \(0.350914\pi\)
\(542\) −1.52786 −0.0656274
\(543\) 22.9443 0.984633
\(544\) 0 0
\(545\) 14.1115 0.604468
\(546\) 6.85410 0.293328
\(547\) 20.8197 0.890184 0.445092 0.895485i \(-0.353171\pi\)
0.445092 + 0.895485i \(0.353171\pi\)
\(548\) 1.41641 0.0605059
\(549\) 26.8328 1.14520
\(550\) −25.1246 −1.07132
\(551\) −20.6525 −0.879825
\(552\) 2.23607 0.0951734
\(553\) 2.76393 0.117534
\(554\) 10.3820 0.441087
\(555\) −9.52786 −0.404435
\(556\) 7.09017 0.300690
\(557\) −31.2361 −1.32351 −0.661757 0.749718i \(-0.730189\pi\)
−0.661757 + 0.749718i \(0.730189\pi\)
\(558\) −29.1246 −1.23294
\(559\) 27.4164 1.15959
\(560\) 6.00000 0.253546
\(561\) 0 0
\(562\) 15.7082 0.662611
\(563\) 39.5967 1.66880 0.834402 0.551156i \(-0.185813\pi\)
0.834402 + 0.551156i \(0.185813\pi\)
\(564\) −3.38197 −0.142406
\(565\) −8.00000 −0.336563
\(566\) −1.70820 −0.0718012
\(567\) −1.00000 −0.0419961
\(568\) −12.8885 −0.540791
\(569\) 14.1803 0.594471 0.297235 0.954804i \(-0.403935\pi\)
0.297235 + 0.954804i \(0.403935\pi\)
\(570\) −5.52786 −0.231537
\(571\) 29.7082 1.24325 0.621625 0.783315i \(-0.286473\pi\)
0.621625 + 0.783315i \(0.286473\pi\)
\(572\) 11.7082 0.489545
\(573\) 1.81966 0.0760174
\(574\) 3.61803 0.151014
\(575\) 3.47214 0.144798
\(576\) −8.47214 −0.353006
\(577\) −28.7082 −1.19514 −0.597569 0.801817i \(-0.703867\pi\)
−0.597569 + 0.801817i \(0.703867\pi\)
\(578\) 27.5066 1.14412
\(579\) 22.4164 0.931594
\(580\) 5.70820 0.237020
\(581\) 2.47214 0.102561
\(582\) −15.7082 −0.651126
\(583\) −30.2492 −1.25279
\(584\) 15.0000 0.620704
\(585\) 10.4721 0.432970
\(586\) 3.70820 0.153184
\(587\) −14.8885 −0.614516 −0.307258 0.951626i \(-0.599411\pi\)
−0.307258 + 0.951626i \(0.599411\pi\)
\(588\) −0.618034 −0.0254873
\(589\) 24.8754 1.02497
\(590\) 20.9443 0.862262
\(591\) 12.5279 0.515327
\(592\) −37.4164 −1.53780
\(593\) −10.3607 −0.425462 −0.212731 0.977111i \(-0.568236\pi\)
−0.212731 + 0.977111i \(0.568236\pi\)
\(594\) 36.1803 1.48450
\(595\) 0 0
\(596\) 6.65248 0.272496
\(597\) 19.8885 0.813984
\(598\) −6.85410 −0.280285
\(599\) 6.47214 0.264444 0.132222 0.991220i \(-0.457789\pi\)
0.132222 + 0.991220i \(0.457789\pi\)
\(600\) 7.76393 0.316961
\(601\) 17.7639 0.724606 0.362303 0.932060i \(-0.381991\pi\)
0.362303 + 0.932060i \(0.381991\pi\)
\(602\) −10.4721 −0.426812
\(603\) −20.3607 −0.829151
\(604\) 4.43769 0.180567
\(605\) 11.1246 0.452280
\(606\) 21.7082 0.881836
\(607\) 29.3050 1.18945 0.594726 0.803929i \(-0.297261\pi\)
0.594726 + 0.803929i \(0.297261\pi\)
\(608\) −9.34752 −0.379092
\(609\) 7.47214 0.302786
\(610\) 26.8328 1.08643
\(611\) −23.1803 −0.937776
\(612\) 0 0
\(613\) 0.180340 0.00728386 0.00364193 0.999993i \(-0.498841\pi\)
0.00364193 + 0.999993i \(0.498841\pi\)
\(614\) −24.9443 −1.00667
\(615\) −2.76393 −0.111452
\(616\) 10.0000 0.402911
\(617\) 30.7639 1.23851 0.619255 0.785190i \(-0.287435\pi\)
0.619255 + 0.785190i \(0.287435\pi\)
\(618\) −3.70820 −0.149166
\(619\) 37.4853 1.50666 0.753331 0.657642i \(-0.228446\pi\)
0.753331 + 0.657642i \(0.228446\pi\)
\(620\) −6.87539 −0.276122
\(621\) −5.00000 −0.200643
\(622\) −53.2148 −2.13372
\(623\) 8.94427 0.358345
\(624\) −20.5623 −0.823151
\(625\) 4.41641 0.176656
\(626\) −11.7082 −0.467954
\(627\) −12.3607 −0.493638
\(628\) −7.05573 −0.281554
\(629\) 0 0
\(630\) −4.00000 −0.159364
\(631\) −23.4164 −0.932192 −0.466096 0.884734i \(-0.654340\pi\)
−0.466096 + 0.884734i \(0.654340\pi\)
\(632\) −6.18034 −0.245841
\(633\) 12.0000 0.476957
\(634\) 50.6525 2.01167
\(635\) 13.2361 0.525257
\(636\) −4.18034 −0.165761
\(637\) −4.23607 −0.167839
\(638\) 54.0689 2.14061
\(639\) 11.5279 0.456035
\(640\) −16.8328 −0.665375
\(641\) 24.9443 0.985240 0.492620 0.870245i \(-0.336039\pi\)
0.492620 + 0.870245i \(0.336039\pi\)
\(642\) 10.9443 0.431936
\(643\) −4.29180 −0.169252 −0.0846260 0.996413i \(-0.526970\pi\)
−0.0846260 + 0.996413i \(0.526970\pi\)
\(644\) 0.618034 0.0243540
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) −27.4721 −1.08004 −0.540021 0.841652i \(-0.681584\pi\)
−0.540021 + 0.841652i \(0.681584\pi\)
\(648\) 2.23607 0.0878410
\(649\) 46.8328 1.83835
\(650\) −23.7984 −0.933449
\(651\) −9.00000 −0.352738
\(652\) −1.96556 −0.0769772
\(653\) 33.4721 1.30987 0.654933 0.755687i \(-0.272697\pi\)
0.654933 + 0.755687i \(0.272697\pi\)
\(654\) 18.4721 0.722318
\(655\) −17.2361 −0.673469
\(656\) −10.8541 −0.423781
\(657\) −13.4164 −0.523424
\(658\) 8.85410 0.345169
\(659\) 11.2361 0.437695 0.218848 0.975759i \(-0.429770\pi\)
0.218848 + 0.975759i \(0.429770\pi\)
\(660\) 3.41641 0.132983
\(661\) −28.2918 −1.10042 −0.550212 0.835025i \(-0.685453\pi\)
−0.550212 + 0.835025i \(0.685453\pi\)
\(662\) 20.5623 0.799177
\(663\) 0 0
\(664\) −5.52786 −0.214523
\(665\) 3.41641 0.132483
\(666\) 24.9443 0.966571
\(667\) −7.47214 −0.289322
\(668\) −8.58359 −0.332109
\(669\) 23.4164 0.905331
\(670\) −20.3607 −0.786602
\(671\) 60.0000 2.31627
\(672\) 3.38197 0.130462
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −51.1246 −1.96925
\(675\) −17.3607 −0.668213
\(676\) 3.05573 0.117528
\(677\) 18.7639 0.721156 0.360578 0.932729i \(-0.382579\pi\)
0.360578 + 0.932729i \(0.382579\pi\)
\(678\) −10.4721 −0.402180
\(679\) 9.70820 0.372567
\(680\) 0 0
\(681\) −29.8885 −1.14533
\(682\) −65.1246 −2.49375
\(683\) 17.1803 0.657387 0.328694 0.944437i \(-0.393392\pi\)
0.328694 + 0.944437i \(0.393392\pi\)
\(684\) 3.41641 0.130630
\(685\) 2.83282 0.108236
\(686\) 1.61803 0.0617768
\(687\) −19.2361 −0.733902
\(688\) 31.4164 1.19774
\(689\) −28.6525 −1.09157
\(690\) −2.00000 −0.0761387
\(691\) −10.8328 −0.412100 −0.206050 0.978541i \(-0.566061\pi\)
−0.206050 + 0.978541i \(0.566061\pi\)
\(692\) −7.70820 −0.293022
\(693\) −8.94427 −0.339765
\(694\) −48.3607 −1.83575
\(695\) 14.1803 0.537891
\(696\) −16.7082 −0.633323
\(697\) 0 0
\(698\) −23.7984 −0.900782
\(699\) 2.52786 0.0956126
\(700\) 2.14590 0.0811073
\(701\) −27.2361 −1.02869 −0.514346 0.857583i \(-0.671965\pi\)
−0.514346 + 0.857583i \(0.671965\pi\)
\(702\) 34.2705 1.29346
\(703\) −21.3050 −0.803531
\(704\) −18.9443 −0.713989
\(705\) −6.76393 −0.254744
\(706\) 10.0902 0.379749
\(707\) −13.4164 −0.504576
\(708\) 6.47214 0.243238
\(709\) −46.2492 −1.73693 −0.868463 0.495754i \(-0.834892\pi\)
−0.868463 + 0.495754i \(0.834892\pi\)
\(710\) 11.5279 0.432633
\(711\) 5.52786 0.207311
\(712\) −20.0000 −0.749532
\(713\) 9.00000 0.337053
\(714\) 0 0
\(715\) 23.4164 0.875724
\(716\) −3.27051 −0.122225
\(717\) 20.2361 0.755730
\(718\) −14.9443 −0.557715
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 12.0000 0.447214
\(721\) 2.29180 0.0853509
\(722\) 18.3820 0.684106
\(723\) −5.41641 −0.201438
\(724\) −14.1803 −0.527008
\(725\) −25.9443 −0.963546
\(726\) 14.5623 0.540458
\(727\) −1.23607 −0.0458432 −0.0229216 0.999737i \(-0.507297\pi\)
−0.0229216 + 0.999737i \(0.507297\pi\)
\(728\) 9.47214 0.351061
\(729\) 13.0000 0.481481
\(730\) −13.4164 −0.496564
\(731\) 0 0
\(732\) 8.29180 0.306474
\(733\) −4.94427 −0.182621 −0.0913104 0.995822i \(-0.529106\pi\)
−0.0913104 + 0.995822i \(0.529106\pi\)
\(734\) 50.8328 1.87627
\(735\) −1.23607 −0.0455931
\(736\) −3.38197 −0.124661
\(737\) −45.5279 −1.67704
\(738\) 7.23607 0.266363
\(739\) −4.70820 −0.173194 −0.0865970 0.996243i \(-0.527599\pi\)
−0.0865970 + 0.996243i \(0.527599\pi\)
\(740\) 5.88854 0.216467
\(741\) −11.7082 −0.430112
\(742\) 10.9443 0.401777
\(743\) −34.4721 −1.26466 −0.632330 0.774699i \(-0.717901\pi\)
−0.632330 + 0.774699i \(0.717901\pi\)
\(744\) 20.1246 0.737804
\(745\) 13.3050 0.487456
\(746\) −42.0689 −1.54025
\(747\) 4.94427 0.180901
\(748\) 0 0
\(749\) −6.76393 −0.247149
\(750\) −16.9443 −0.618717
\(751\) −35.4164 −1.29236 −0.646182 0.763184i \(-0.723635\pi\)
−0.646182 + 0.763184i \(0.723635\pi\)
\(752\) −26.5623 −0.968628
\(753\) −21.7082 −0.791091
\(754\) 51.2148 1.86513
\(755\) 8.87539 0.323008
\(756\) −3.09017 −0.112388
\(757\) 24.8328 0.902564 0.451282 0.892381i \(-0.350967\pi\)
0.451282 + 0.892381i \(0.350967\pi\)
\(758\) −35.1246 −1.27578
\(759\) −4.47214 −0.162328
\(760\) −7.63932 −0.277107
\(761\) −0.347524 −0.0125977 −0.00629887 0.999980i \(-0.502005\pi\)
−0.00629887 + 0.999980i \(0.502005\pi\)
\(762\) 17.3262 0.627663
\(763\) −11.4164 −0.413302
\(764\) −1.12461 −0.0406870
\(765\) 0 0
\(766\) 3.70820 0.133983
\(767\) 44.3607 1.60177
\(768\) −13.5623 −0.489388
\(769\) −38.7639 −1.39786 −0.698932 0.715189i \(-0.746341\pi\)
−0.698932 + 0.715189i \(0.746341\pi\)
\(770\) −8.94427 −0.322329
\(771\) 0.236068 0.00850178
\(772\) −13.8541 −0.498620
\(773\) 9.34752 0.336207 0.168104 0.985769i \(-0.446236\pi\)
0.168104 + 0.985769i \(0.446236\pi\)
\(774\) −20.9443 −0.752826
\(775\) 31.2492 1.12251
\(776\) −21.7082 −0.779279
\(777\) 7.70820 0.276530
\(778\) −31.4164 −1.12633
\(779\) −6.18034 −0.221434
\(780\) 3.23607 0.115870
\(781\) 25.7771 0.922377
\(782\) 0 0
\(783\) 37.3607 1.33516
\(784\) −4.85410 −0.173361
\(785\) −14.1115 −0.503659
\(786\) −22.5623 −0.804771
\(787\) 44.3607 1.58129 0.790644 0.612276i \(-0.209746\pi\)
0.790644 + 0.612276i \(0.209746\pi\)
\(788\) −7.74265 −0.275820
\(789\) 9.05573 0.322392
\(790\) 5.52786 0.196673
\(791\) 6.47214 0.230123
\(792\) 20.0000 0.710669
\(793\) 56.8328 2.01819
\(794\) −9.14590 −0.324576
\(795\) −8.36068 −0.296523
\(796\) −12.2918 −0.435671
\(797\) −41.2361 −1.46066 −0.730328 0.683096i \(-0.760633\pi\)
−0.730328 + 0.683096i \(0.760633\pi\)
\(798\) 4.47214 0.158312
\(799\) 0 0
\(800\) −11.7426 −0.415165
\(801\) 17.8885 0.632061
\(802\) −25.7082 −0.907788
\(803\) −30.0000 −1.05868
\(804\) −6.29180 −0.221895
\(805\) 1.23607 0.0435657
\(806\) −61.6869 −2.17283
\(807\) −13.1803 −0.463970
\(808\) 30.0000 1.05540
\(809\) 20.8328 0.732443 0.366221 0.930528i \(-0.380651\pi\)
0.366221 + 0.930528i \(0.380651\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −0.639320 −0.0224496 −0.0112248 0.999937i \(-0.503573\pi\)
−0.0112248 + 0.999937i \(0.503573\pi\)
\(812\) −4.61803 −0.162061
\(813\) −0.944272 −0.0331171
\(814\) 55.7771 1.95499
\(815\) −3.93112 −0.137701
\(816\) 0 0
\(817\) 17.8885 0.625841
\(818\) 58.4508 2.04369
\(819\) −8.47214 −0.296040
\(820\) 1.70820 0.0596531
\(821\) 49.4164 1.72464 0.862322 0.506360i \(-0.169009\pi\)
0.862322 + 0.506360i \(0.169009\pi\)
\(822\) 3.70820 0.129338
\(823\) −41.5410 −1.44803 −0.724014 0.689785i \(-0.757705\pi\)
−0.724014 + 0.689785i \(0.757705\pi\)
\(824\) −5.12461 −0.178524
\(825\) −15.5279 −0.540611
\(826\) −16.9443 −0.589567
\(827\) 45.3050 1.57541 0.787704 0.616054i \(-0.211270\pi\)
0.787704 + 0.616054i \(0.211270\pi\)
\(828\) 1.23607 0.0429563
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 4.94427 0.171618
\(831\) 6.41641 0.222583
\(832\) −17.9443 −0.622106
\(833\) 0 0
\(834\) 18.5623 0.642760
\(835\) −17.1672 −0.594095
\(836\) 7.63932 0.264211
\(837\) −45.0000 −1.55543
\(838\) −28.3607 −0.979703
\(839\) −38.0689 −1.31428 −0.657142 0.753767i \(-0.728235\pi\)
−0.657142 + 0.753767i \(0.728235\pi\)
\(840\) 2.76393 0.0953647
\(841\) 26.8328 0.925270
\(842\) 57.3050 1.97486
\(843\) 9.70820 0.334368
\(844\) −7.41641 −0.255283
\(845\) 6.11146 0.210240
\(846\) 17.7082 0.608821
\(847\) −9.00000 −0.309244
\(848\) −32.8328 −1.12748
\(849\) −1.05573 −0.0362325
\(850\) 0 0
\(851\) −7.70820 −0.264234
\(852\) 3.56231 0.122043
\(853\) −8.83282 −0.302430 −0.151215 0.988501i \(-0.548319\pi\)
−0.151215 + 0.988501i \(0.548319\pi\)
\(854\) −21.7082 −0.742839
\(855\) 6.83282 0.233677
\(856\) 15.1246 0.516949
\(857\) −57.5410 −1.96556 −0.982782 0.184769i \(-0.940846\pi\)
−0.982782 + 0.184769i \(0.940846\pi\)
\(858\) 30.6525 1.04646
\(859\) 38.4164 1.31075 0.655375 0.755303i \(-0.272510\pi\)
0.655375 + 0.755303i \(0.272510\pi\)
\(860\) −4.94427 −0.168598
\(861\) 2.23607 0.0762050
\(862\) 6.76393 0.230380
\(863\) −2.23607 −0.0761166 −0.0380583 0.999276i \(-0.512117\pi\)
−0.0380583 + 0.999276i \(0.512117\pi\)
\(864\) 16.9098 0.575284
\(865\) −15.4164 −0.524174
\(866\) −22.6525 −0.769762
\(867\) 17.0000 0.577350
\(868\) 5.56231 0.188797
\(869\) 12.3607 0.419307
\(870\) 14.9443 0.506658
\(871\) −43.1246 −1.46122
\(872\) 25.5279 0.864483
\(873\) 19.4164 0.657146
\(874\) −4.47214 −0.151272
\(875\) 10.4721 0.354023
\(876\) −4.14590 −0.140077
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) 52.2705 1.76404
\(879\) 2.29180 0.0773004
\(880\) 26.8328 0.904534
\(881\) −23.5967 −0.794995 −0.397497 0.917603i \(-0.630121\pi\)
−0.397497 + 0.917603i \(0.630121\pi\)
\(882\) 3.23607 0.108964
\(883\) −25.8885 −0.871219 −0.435609 0.900136i \(-0.643467\pi\)
−0.435609 + 0.900136i \(0.643467\pi\)
\(884\) 0 0
\(885\) 12.9443 0.435117
\(886\) −24.5623 −0.825187
\(887\) −18.8885 −0.634215 −0.317108 0.948390i \(-0.602712\pi\)
−0.317108 + 0.948390i \(0.602712\pi\)
\(888\) −17.2361 −0.578405
\(889\) −10.7082 −0.359142
\(890\) 17.8885 0.599625
\(891\) −4.47214 −0.149822
\(892\) −14.4721 −0.484563
\(893\) −15.1246 −0.506126
\(894\) 17.4164 0.582492
\(895\) −6.54102 −0.218642
\(896\) 13.6180 0.454947
\(897\) −4.23607 −0.141438
\(898\) 33.7082 1.12486
\(899\) −67.2492 −2.24289
\(900\) 4.29180 0.143060
\(901\) 0 0
\(902\) 16.1803 0.538746
\(903\) −6.47214 −0.215379
\(904\) −14.4721 −0.481336
\(905\) −28.3607 −0.942741
\(906\) 11.6180 0.385983
\(907\) −54.5410 −1.81100 −0.905502 0.424341i \(-0.860506\pi\)
−0.905502 + 0.424341i \(0.860506\pi\)
\(908\) 18.4721 0.613019
\(909\) −26.8328 −0.889988
\(910\) −8.47214 −0.280849
\(911\) 49.7771 1.64919 0.824594 0.565725i \(-0.191403\pi\)
0.824594 + 0.565725i \(0.191403\pi\)
\(912\) −13.4164 −0.444262
\(913\) 11.0557 0.365891
\(914\) −8.94427 −0.295850
\(915\) 16.5836 0.548237
\(916\) 11.8885 0.392809
\(917\) 13.9443 0.460480
\(918\) 0 0
\(919\) −9.70820 −0.320244 −0.160122 0.987097i \(-0.551189\pi\)
−0.160122 + 0.987097i \(0.551189\pi\)
\(920\) −2.76393 −0.0911241
\(921\) −15.4164 −0.507988
\(922\) 48.9230 1.61119
\(923\) 24.4164 0.803676
\(924\) −2.76393 −0.0909267
\(925\) −26.7639 −0.879993
\(926\) 0 0
\(927\) 4.58359 0.150545
\(928\) 25.2705 0.829545
\(929\) 20.3475 0.667581 0.333790 0.942647i \(-0.391672\pi\)
0.333790 + 0.942647i \(0.391672\pi\)
\(930\) −18.0000 −0.590243
\(931\) −2.76393 −0.0905842
\(932\) −1.56231 −0.0511750
\(933\) −32.8885 −1.07672
\(934\) 33.1246 1.08387
\(935\) 0 0
\(936\) 18.9443 0.619213
\(937\) −16.8754 −0.551295 −0.275647 0.961259i \(-0.588892\pi\)
−0.275647 + 0.961259i \(0.588892\pi\)
\(938\) 16.4721 0.537834
\(939\) −7.23607 −0.236140
\(940\) 4.18034 0.136348
\(941\) −23.0132 −0.750207 −0.375104 0.926983i \(-0.622393\pi\)
−0.375104 + 0.926983i \(0.622393\pi\)
\(942\) −18.4721 −0.601855
\(943\) −2.23607 −0.0728164
\(944\) 50.8328 1.65447
\(945\) −6.18034 −0.201046
\(946\) −46.8328 −1.52267
\(947\) 26.5967 0.864278 0.432139 0.901807i \(-0.357759\pi\)
0.432139 + 0.901807i \(0.357759\pi\)
\(948\) 1.70820 0.0554799
\(949\) −28.4164 −0.922436
\(950\) −15.5279 −0.503790
\(951\) 31.3050 1.01513
\(952\) 0 0
\(953\) 47.8885 1.55126 0.775631 0.631187i \(-0.217432\pi\)
0.775631 + 0.631187i \(0.217432\pi\)
\(954\) 21.8885 0.708668
\(955\) −2.24922 −0.0727832
\(956\) −12.5066 −0.404492
\(957\) 33.4164 1.08020
\(958\) −27.7082 −0.895211
\(959\) −2.29180 −0.0740060
\(960\) −5.23607 −0.168993
\(961\) 50.0000 1.61290
\(962\) 52.8328 1.70340
\(963\) −13.5279 −0.435929
\(964\) 3.34752 0.107816
\(965\) −27.7082 −0.891959
\(966\) 1.61803 0.0520594
\(967\) 27.0689 0.870477 0.435238 0.900315i \(-0.356664\pi\)
0.435238 + 0.900315i \(0.356664\pi\)
\(968\) 20.1246 0.646830
\(969\) 0 0
\(970\) 19.4164 0.623423
\(971\) 20.2918 0.651195 0.325597 0.945509i \(-0.394435\pi\)
0.325597 + 0.945509i \(0.394435\pi\)
\(972\) −9.88854 −0.317175
\(973\) −11.4721 −0.367780
\(974\) 32.5623 1.04336
\(975\) −14.7082 −0.471040
\(976\) 65.1246 2.08459
\(977\) 12.6525 0.404789 0.202394 0.979304i \(-0.435128\pi\)
0.202394 + 0.979304i \(0.435128\pi\)
\(978\) −5.14590 −0.164548
\(979\) 40.0000 1.27841
\(980\) 0.763932 0.0244029
\(981\) −22.8328 −0.728996
\(982\) −26.2705 −0.838326
\(983\) 18.1803 0.579863 0.289931 0.957047i \(-0.406368\pi\)
0.289931 + 0.957047i \(0.406368\pi\)
\(984\) −5.00000 −0.159394
\(985\) −15.4853 −0.493402
\(986\) 0 0
\(987\) 5.47214 0.174180
\(988\) 7.23607 0.230210
\(989\) 6.47214 0.205802
\(990\) −17.8885 −0.568535
\(991\) −31.0557 −0.986518 −0.493259 0.869883i \(-0.664194\pi\)
−0.493259 + 0.869883i \(0.664194\pi\)
\(992\) −30.4377 −0.966398
\(993\) 12.7082 0.403283
\(994\) −9.32624 −0.295810
\(995\) −24.5836 −0.779352
\(996\) 1.52786 0.0484122
\(997\) 35.3050 1.11812 0.559060 0.829128i \(-0.311162\pi\)
0.559060 + 0.829128i \(0.311162\pi\)
\(998\) 37.6869 1.19296
\(999\) 38.5410 1.21938
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 161.2.a.b.1.1 2
3.2 odd 2 1449.2.a.i.1.2 2
4.3 odd 2 2576.2.a.s.1.2 2
5.4 even 2 4025.2.a.i.1.2 2
7.6 odd 2 1127.2.a.d.1.1 2
23.22 odd 2 3703.2.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.b.1.1 2 1.1 even 1 trivial
1127.2.a.d.1.1 2 7.6 odd 2
1449.2.a.i.1.2 2 3.2 odd 2
2576.2.a.s.1.2 2 4.3 odd 2
3703.2.a.b.1.1 2 23.22 odd 2
4025.2.a.i.1.2 2 5.4 even 2